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On the functoriality of marked families. (English) Zbl 1348.14010

It is natural to use Gröbner bases with respect to a term ordering of variables to study the Hilbert scheme \(\mathbf{ Hilb}^{p(t)}\) parametrizing closed subschemes \(X \subset \mathbb P^n_k\) with Hilbert polynomial \(p(t)\). The Gröbner strata are the sets of homogeneous ideals having a fixed monomial ideal as initial ideal. These strata come with a scheme structure, which one can compare with the scheme structure on the Hilbert scheme. M. Lederer proved that for Hilbert schemes of zero-dimensional subschemes in \(\mathbb A^n_k\), the Gröbner strata are locally closed in \(\mathbf{ Hilb}^{p(t)}\) [J. Commut. Alg. 3, 349–404 (2011; Zbl 1237.14012)].
Earlier work of the authors shows that Gröbner strata do not in general form an open cover of \(\mathbf{ Hilb}^{p(t)}\) [Rend. Semin. Mat. Univ. Padova 126, 11–45 (2011; Zbl 1236.14006)], so instead they use the marked families seen in work of F. Cioffi and M. Roggero [J. Symb. Comput. 46, No. 9, 1070–1084 (2011; Zbl 1231.13024)] and C. Bertone et al. [J. Symb. Comput. 50, 263–290 (2013; Zbl 1314.14008)]. To describe them, let \(J \subset {\mathbb Z}[x_0, \dots, x_n]\) be a strongly stable ideal, i.e. \(J\) is monomial and for each \(x^{\alpha} \in J\), \(x_j | x^{\alpha}\) and \(x_i > x_j\) imply \((x_i/x_j) x^{\alpha} \in J\). A strongly stable ideal \(J\) defines a functor \(\underline{\mathbf{ Mf}}_J\) from noetherian rings to sets via \[ A \mapsto \{ \text{ideals } I \subset A[x_0, \dots, x_n]: A[x_0, \dots, x_n] = I \oplus \langle {\mathcal N} (J) \rangle \} \] where \({\mathcal N} (J) = \{ x^{\alpha}: x^{\alpha} \not \in J \}\), which can be characterized in terms of marked families. The authors show that \(\underline{\mathbf{ Mf}}_J\) is a Zariski sheaf explicitly construct a scheme \(\mathbf{ Mf}_J\) representing it. For \(J\) a saturated strongly stable ideal, they show that \(\mathbf{ Mf}_{J_{\geq s}}\) is locally closed in \(\mathbf{ Hilb}^{p(t)}\) for arbitrary \(s\) and open in \(\mathbf{ Hilb}^{p(t)}\) for sufficiently large \(s\). Since the schemes representing Gröbner strata are closed in \(\mathbf{ Mf}\), they extend Lederer’s result above to Hilbert polynomials of any degree. These ideas were used by J. Brachat et al. [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 16, No. 1, 65–96 (2016; Zbl 1342.14008)]. The paper contains many computational examples, including a detailed study of the Hilbert scheme of zero-dimensional subschemes in \(\mathbb P^3\) of degree \(7\).

MSC:

14C05 Parametrization (Chow and Hilbert schemes)
13P99 Computational aspects and applications of commutative rings
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References:

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